This Document Contains Chapters 5 to 6 Chapter 5 How to Value Bonds and Shares 1. The main characteristics of a bond are a) the face value or principal, b) the term to maturity, c) the coupon rate, and d) the frequency of coupon payments. A zero coupon bond has no coupon and only pays the face value at expiry. A coupon bond can have a fixed coupon where coupon payments are fixed for the life of a bond or a floating coupon, where coupon payments are based on some benchmark rate such as LIBOR or EURIBOR. 2. To value any bond, you should use the present value formula: A zero coupon bond has no coupon and so C = 0 in the above formula. A floating rate bond will use the same formula except that C will change each period and needs to be estimated. To accommodate this into the PV formula, you would use expected cash flows: 3. A coupon rate is the periodic interest (as a percentage of face value) paid by a bond and yield to maturity is the return that a bondholder receives from the bond. The coupon rate and yield to maturity can be different. For example, when the coupon rate is greater than the yield to maturity, the bond will be priced above face value. When the coupon rate is less than the yield to maturity, the bond will be priced below face value. When the coupon rate is equal to yield to maturity, the bond price will be equal to the face value. 4. The value of any investment depends on the present value of its cash flows; i.e., what investors will actually receive. The cash flows from equities are the dividends and any change in the value of the shares (capital gains). 5. The general method for valuing an equity is to find the present value of all expected future dividends. The dividend growth model presented in the text is only valid (i) if dividends are expected to occur forever; that is, the equity provides dividends in perpetuity, and (ii) if a constant growth rate of dividends occurs forever. A violation of the first assumption might be a company that is expected to cease operations and dissolve itself some finite number of years from now. The shares of such a company would be valued by applying the general method of valuation explained in this chapter. A violation of the second assumption might be a start-up firm that isn’t currently paying any dividends, but is expected to eventually start making dividend payments some number of years from now. This equity would also be valued by the general dividend valuation method explained in this chapter. 6. Yes. If the dividend grows at a steady rate, so does the share price. In other words, the dividend growth rate and the capital gains yield are the same. 2 T T Principal PV ... 1 (1 ) (1 ) (1 ) C C C = +R + +R + + + R + + R 1 2 2 T T [ ] [ ] [ ] Principal PV ... 1 (1 ) (1 ) (1 ) E C E C E CT = +R + +R + + + R + + R This Document Contains Chapters 5 to 6 7. NPVGO means the Net Present Value of Growth Opportunities and this is a measure of the value of a firm resulting from future possible investments. It is equivalent to the dividend growth model but tends to be used to analyse the value arising from a possible new investment rather than continual growth. 8. The three factors are: 1) The company’s future growth opportunities. 2) The company’s level of risk, which determines the interest rate used to discount cash flows. 3) The accounting method used. 9. The formulae presented in this chapter require the analyst to estimate future cash flows from the bond or equity, and risk of each investment. Investors use all manners of information to estimate these. Information websites simply provide the information that many find relevant in the valuation decision. 10. a. P = €41.25 x A10 0.04 + €1000/ (1.04)10 P = €1010.14 b. P = €41.25 x A10 0.05 + €1000/ (1.05)10 P = €932.43 c. P = €41.25 x A10 0.1 + €1000/ (1.1)10 P = €639.01 11. Quarterly coupon interest rate = 0.02/4 = 0.005 €9,984.50 = €5 x A6 R/4 + 1000/ (1+R/4)6 Yield to maturity is 2.105% 12. a. Current price at a constant growth of 5 per cent per year indefinitely P = Div1/ (R – g), where as Div1 = D0 (1+g) i.e. Div1 = €2 (1.05) = €2.1 P = €2.1/ (0.11 – 0.05) = €35 b. Price in 3 years P = P0(1+g)3 = €35(1.05)3 = €40.52 c. Price in 15 years P = P0(1+g)15 = €35(1.05)15 = €72.76 13. Div0 = €.013; g = 7%; P0 = €.52 Div1 = €.013(1+.07) = €.01391 r = Div1/P0 + g = .013/.52 +.07 = 9.675% The required return is 9.675%. In reality, most investors would be concerned at the financial difficulties that the firm has faced and why the share price dropped by 20 percent. Has the company remedied the issues or are they expected to continue? Given the risk of the company, investors would likely expect a higher return from their investment. 14. The retention ratio of Severn Trent plc is (£1.01 - .67)/£1.01 = 0.3366 The dividends are given below: Year 1 2 3 4 5 Dividend £.67 £.6968 £.72467 £.75366 £.78381 Earnings at year 5 are £.78381/(1-0.3366) = £1.1816. To value this company, we discount the dividend stream to present day and then treat the £1.1816 as a perpetuity since the firm will no longer grow. The present value of the dividend stream from years 1 to 4 is £2.295 when discounted at a rate of 9%. We use the perpetuity shortcut to value the remaining cash flows. V4 = £1.1816/.09 = £13.1284 V0 = £13.1284/(1.094) = £9.30 So the share price of Severn Trent is £2.295 + £9.300 = £11.59. 15. If Severn Trent pays out all its earnings as dividends, the value of the company today is £.67/.09 = £7.44. This means that investors pay £11.59 - £7.44 = £4.1507 for the growth opportunities. 16. Share price P = Div x A60.1 = €10 x 4.3553 = €43.55 17. Price of ABN AMRO's bond =€41.25 PVIFA (10%, 10) + €1,000 PVIF (10%, 10) =€253.46 + €385.54 = €639.01 The price of the RCI Banque's bond €9,984.50= €50 PVIFA (R/4, 6) + €10,000 PVIF(R/4, 6) YTM = R = 2.105% If YTM remains unchanged, the price of ABN AMRO's bond one year from now would be: P1 =€41.25 PVIFA (10%, 9) +€1,000 PVIF (10%, 9) =€237.56 +€424.10 =€661.66 If YTM remains unchanged, the price of ABN AMRO's bond two years from now would be: P2 =€41.25 PVIFA (10%, 8) + €1,000 PVIF (10%, 8) =€220.07 + €466.51 =€686.57 If YTM remains unchanged, the price of RCI Banque's bond one year from now would be: P1 = €50 PVIFA (2.105%, 2) + €10,000 PVIF (2.105%, 2) =€99.22 + €9895.56 = €9,994.78 As the maturity period for RCI Banque's bond is 18 months, we cannot calculate the bond price two years from now. All other things remaining constant, as a bond approaches its maturity, its price approaches its par value. This is called “pull to par.” 18. Price, P = 50 x A6 0.03 + 1000/ (1+R)6 = 1,108.34 One year later, P = 50 x A5 0.03 + 1000/ (1+R)5 = 1,091.59 If yields fall to 2%: P = 50 x A5 0.02 + 1000/ (1+R)5 = 1,141.40 The return on the bond is [(P1 + C)/P0] – 1 In the first example, the return is [(1,091.59 + 50)/1,108.34] – 1 = 3% In the second example, the return is [(1,141.40 +50)/1,108.34] – 1 = 7.49% 19. Here we have an equity that pays no dividends for 9 years. Once it begins paying dividends, it will have a constant growth rate of dividends. We can use the constant growth model at that point. It is important to remember that general form of the constant dividend growth formula is: Pt = [Dt × (1 + g)] / (R – g) This means that since we will use the dividend in Year 10, we will be finding the share price in Year 9. The dividend growth model is similar to the PVA and the PV of a perpetuity: The equation gives you the PV one period before the first payment. So, the price of the equity in Year 9 will be: P9 = D10 / (R – g) = £8.00 / (.13 – .06) = £114.29 The price of the share today is simply the PV of the share price in the future. We simply discount the future share price at the required return. The price of the share today will be: P0 = £114.29 / 1.139 = £38.04 20. The price of a preference share is the dividend payment divided by the required return. We know the dividend payment in Year 4, so we can find the price of the share in Year 3, one year before the first dividend payment. Doing so, we get: P3 = £5.00 / .08 = £62.50 The price of the share today is the PV of the share price in the future, so the price today will be: P0 = £62.50 / (1.08)3 = £49.61 21. First we must calculate the earnings per share, dividend and book value. Below is given the range of values for each variable. 1 2 3 4 5 Book Value €50 €53.50 €57.25 €61.25 €65.54 ROE 14% 14% 14% 14% 11.50% Payout Ratio 0.5 0.5 0.5 0.5 0.8 EPS €7.00 €7.49 €8.01 €8.58 €7.54 Dividend €3.5 €3.745 €4.007 €4.288 €6.03 As an example, EPS is calculated by multiplying ROE by Book Value, and Dividend is calculated by multiplying EPS by the payout ratio. To value the shares we must discount the dividends back to time 0. We must also calculate present value of the perpetuity starting in year 5. End of Year Dividend 1 2 3 4 Div1/ (1 + r) €3.5 / (1.115) = €3.139 Div2 / (1 + r)2 €3.745 / (1.115)2 = €3.012 Div3 / (1 + r)3 €4.007 / (1.115)3 = €2.891 Div4 / (1 + r)4 €4.288 / (1.115)4 = €2.774 The growth rate from year 5 is : g = Retention ratio × Return on retained earnings = .2 x .115 = 0.023 V4 = €6.03/(.115 - .023) = €65.54 V0 = €65.54/(1+ .115)4 = €42.4 P0 = €3.139 + €3.012 + €2.891 + €2.774 + €42.4 = €54.22 22. This is, in effect, two growing perpetuity streams. The first stream is the 75% payout and the second stream is the 25% payout. We should value these separately and then discount back to zero. The perpetuity streams are as follows: 0.5 1 1.5 2 2.5 3 3.5 Annual Dividend €1.112 4 €1.14577 2 €1.18014 5 interim payment €0.834 3 €0.859 3 €0.88510 9 final payment €0.278 1 €0.28644 3 €0.29503 6 Consider the interim payment stream first. We can value this at t = .5 as: V0.5 = Div0.5 + Div1.5/(.08 - .03) = €0.8343 + €0.8593/0.05 = €18.02 The six-month discount rate is R = (1.08)1/2 – 1 = 3.923% V0 = €18.02/(1.03923) = €17.3406 Now consider the final payment stream. V0 = €0.2781/(.08 - .03) = €5.562 P0 = €17.3406 + €5.562 = €22.90 23. The required return of a share consists of two components, the capital gains yield and the dividend yield. In the constant dividend growth model (growing perpetuity equation), the capital gains yield is the same as the dividend growth rate, or algebraically: R = D1/P0 + g We can find the dividend growth rate by the growth rate equation, or: g = ROE × b g = .12 × .75 g = .09 or 9% This is also the growth rate in dividends. To find the current dividend, we can use the information provided about the net income, shares outstanding, and payout ratio. The total dividends paid is the net income times the payout ratio. To find the dividend per share, we can divide the total dividends paid by the number of shares outstanding. So: Dividend per share = (Net income × Payout ratio) / Shares outstanding Dividend per share = (€68,000,000 × .25) / 1,250,000 Dividend per share = €13.60 Now we can use the initial equation for the required return. We must remember that the equation uses the dividend in one year, so: R = D1/P0 + g R = €13.6(1 + .09)/€272 + .09 R = .1445 or 14.45% 24. a. We can find the price of the all the outstanding company equity by using the dividends the same way we would value an individual share. Since earnings are equal to dividends, and there is no growth, the value of the company’s shares today is the present value of a perpetuity, so: P = D / R P = £1,800,000 / .12 P = £15,000,000 The price-earnings ratio is the share price divided by the current earnings, so the price- earnings ratio of each company with no growth is: P/E = Price / Earnings P/E = £15,000,000 / £1,800,000 P/E = 8.33 times b. Since the earnings have increased, the price of the shares will increase. The new price of the all the outstanding company equity is: P = D / R P = (£1,800,000 + £200,000) / .12 P = £16,666,667 The price-earnings ratio is the share price divided by the current earnings, so the price- earnings with the increased earnings is: P/E = Price / Earnings P/E = £16,666,667 / £1,800,000 P/E = 9.26 times c. Since the earnings have increased, the share price will increase. The new price of the all the outstanding company equity is: P = D / R P = (£1,800,000 + £400,000) / .12 P = £18,333,333 The price-earnings ratio is the share price divided by the current earnings, so the price- earnings with the increased earnings is: P/E = Price / Earnings P/E = £18,333,333 / £1,800,000 P/E = 10.19 times 25. a. If the company does not make any new investments, the share price will be the present value of the constant perpetual dividends. In this case, all earnings are paid dividends, so, applying the perpetuity equation, we get: P = Dividend / R P = €4 / .12 P = €33.33 b. The investment is a one-time investment that creates an increase in EPS for two years. To calculate the new share price, we need the cash cow price plus the NPVGO. In this case, the NPVGO is simply the present value of the investment plus the present value of the increases in EPS. So, the NPVGO will be: NPVGO = C1 / (1 + R) + C2 / (1 + R)2 + C3 / (1 + R)3 NPVGO = –€1 / 1.12 + €1.90 / 1.122 + €2.10 / 1.123 NPVGO = €2.12 So, the share price if the company undertakes the investment opportunity will be: P = €33.33 + €2.12 P = €35.45 c. After the project is over, and the earnings increase no longer exists, the share price will revert back to €33.33, the value of the company as a cash cow. 26. a. If the company continues its current operations, it will not grow, so we can value the company as a cash cow. The total value of the company as a cash cow is the present value of the future earnings, which are a perpetuity, so: Cash cow value of company = C / R Cash cow value of company = £60,000,000 / .12 Cash cow value of company = £500,000,000 The value per share is the total value of the company divided by the shares outstanding, so: Share price = £500,000,000 / 10,000,000 Share price = £50 b. To find the value of the investment, we need to find the NPV of the growth opportunities. The initial cash flow occurs today, so it does not need to be discounted. The earnings growth is a perpetuity. Using the present value of a perpetuity equation will give us the value of the earnings growth one period from today, so we need to discount this back to today. The NPVGO of the investment opportunity is: NPVGO = C0 + C1 + (C2 / R) / (1 + R) NPVGO = –£10,000,000 – £12,000,000/1.12 + (£15,000,000 / .12) / (1 + .12) NPVGO = £90,892,857.14 c. The price of a share of equity is the cash cow value plus the NPVGO. We have already calculated the NPVGO for the entire project, so we need to find the NPVGO on a per share basis. The NPVGO on a per share basis is the NPVGO of the project divided by the shares outstanding, which is: NPVGO per share = £90,892,857.14 / 10,000,000 NPVGO per share = £9.09 This means the share price if the company undertakes the project is: Share price = Cash cow price + NPVGO per share Share price = £50 + £9.09 Share price = £59.09 27. First you need to calculate the Free Cash Flow to the Firm. Since interest is presented in the section Cash Flow from Operating Activities, you should use: FCFF = Cash Flow from Operations + Cash Flow from Investing Activities + Net Interest Payment*(1-Tax Rate) FCFF = £354.9 million - £90.2 million + £1.9 million*(1-.23) FCFF = £266.16 million. R = 15%; g = 6%; Value = FCFF/(r-g) = £266.16 million/(.15-.06) = £2,957.37 million. 28. a. If the company does not make any new investments, the share price will be the present value of the constant perpetual dividends. In this case, all earnings are paid as dividends, so, applying the perpetuity equation, we get: P = Dividend / R P = SEK6 / .14 P = SEK42.86 b. The investment occurs every year in the growth opportunity, so the opportunity is a growing perpetuity. So, we first need to find the growth rate. The growth rate is: g = Retention Ratio Return on Retained Earnings g = 0.25 × 0.40 g = 0.10 or 10% Next, we need to calculate the NPV of the investment. During year 3, twenty-five percent of the earnings will be reinvested. Therefore, SEK1.5 is invested (SEK6 .25). One year later, the shareholders receive a 40 percent return on the investment, or SEK0.60 (SEK1.5 × .40), in perpetuity. The perpetuity formula values that stream as of year 3. Since the investment opportunity will continue indefinitely and grows at 10 per cent, apply the growing perpetuity formula to calculate the NPV of the investment as of year 2. Discount that value back two years to today. NPVGO = [(Investment + Return / R) / (R – g)] / (1 + R)2 NPVGO = [(–SEK1.5 + SEK0.60 / .14) / (0.14 – 0.1)] / (1.14)2 NPVGO = SEK53.59 The value of the equity is the PV of the firm without making the investment plus the NPV of the investment, or: P = PV(EPS) + NPVGO P = SEK42.86 + SEK53.58 P = SEK96.45 29. Net Income = 4,456 crore EPS = 73.56 crore Number of Shares = Net Income/EPS = 60.5764 ROE = 17.83% Payout Ratio = 23.7% = .237 Retention Ratio = 1 - .237 = 0.763 g = Retention Ratio x ROE = .763 x 17.83% = 13.60% Div0 = 23.7% x 73.56 crore = 17.4337 crore Div1 = Div0 (1+g) = 17.4337 crore x (1+13.60%) = 19.8055 crore R = 18% P0 = Div1/(r-g) = 19.8055 crore/(.18-.1360) = 450.5632 crore VE = P0 x Number of Shares = 450.5632 crore x 60.5764 = 27,293 crore D/E = 1.31 VD = 1.31 x VE = 1.31 x 27,293 crore = 35,754.48 crore. VFirm = 27,293 crore + 35,754.48 crore = 63,047 crore. 30. To find the capital gains yield and the current yield, we need to find the price of the bond. The current price of Bond P and the price of Bond P in one year is: P: P0 = £110(PVIFA8%,5) + £1,000(PVIF8%,5) = £1,119.78 P1 = £110(PVIFA8%,4) + £1,000(PVIF8%,4) = £1,099.36 Current yield = £110 / £1,119.78 = 9.82% The capital gains yield is: Capital gains yield = (New price – Original price) / Original price Capital gains yield = (£1,099.36 – £1,119.78) / £1,119.78 = –1.82% The current price of Bond D and the price of Bond D in one year is: D: P0 = £50(PVIFA8%,5) + £1,000(PVIF8%,5) = £880.22 P1 = £50(PVIFA8%,4) + £1,000(PVIF8%,4) = £900.64 Current yield = £50 / £880.22 = 5.68% Capital gains yield = (£900.64 – £880.22) / £880.22 = +2.32% All else held constant, premium bonds pay high current income while having price depreciation as maturity nears; discount bonds do not pay high current income but have price appreciation as maturity nears. For either bond, the total return is still 8%, but this return is distributed differently between current income and capital gains. 31. a. The rate of return you expect to earn if you purchase a bond and hold it until maturity is the YTM. The bond price equation for this bond is: P0 = €1,150 = €80(PVIFAR%,10) + €1,000(PVIF R%,10) Using a spreadsheet, financial calculator, or trial and error we find: R = YTM = 5.97% b. To find our HPY, we need to find the price of the bond in two years. The price of the bond in two years, at the new YTM, will be: P2 = €80(PVIFA4.97%,8) + €1,000(PVIF4.97%,8) = €1,196.41 To calculate the HPY, we need to find the interest rate that equates the price we paid for the bond with the cash flows we received. The cash flows we received were €80 each year for two years, and the price of the bond when we sold it. The equation to find our HPY is: P0 = €1,150 = €80(PVIFAR%,2) + €1,196.41(PVIFR%,2) Solving for R, we get: R = HPY = 8.89% The realized HPY is greater than the expected YTM when the bond was bought because YTM dropped by 1 percent; bond prices rise when yields fall. 32. DPS = €2; EPS = €4.62. This means that the retention rate, b, is: b = 1 – (DPS/EPS) = 1 – (€2/€4.62) = .5671 The growth rate: g = b x ROE = .5671 x .1185 = .0672 The discount rate: R = D/P + g = 2/99.63 + .0672 = .08728 = 8.728% 33. The price of any bond (or financial instrument) is the PV of the future cash flows. Even though Bond M makes different coupons payments, to find the price of the bond, we just find the PV of the cash flows. The PV of the cash flows for Bond M is: PM = £1,200(PVIFA5%,16)(PVIF5%,12) + £1,500(PVIFA5%,12)(PVIF5%,28) + £20,000(PVIF5%,40) PM = £13,474.20 Notice that for the coupon payments of £1,500, we found the PVA for the coupon payments, and then discounted the lump sum back to today. Bond N is a zero coupon bond with a £20,000 par value; therefore, the price of the bond is the PV of the par, or: PN = £20,000(PVIF5%,40) = £2,840.91 34. We are asked to find the dividend yield and capital gains yield for each of the equities. All of the equities have a 15 percent required return, which is the sum of the dividend yield and the capital gains yield. To find the components of the total return, we need to find the share price for each equity. Using this share price and the dividend, we can calculate the dividend yield. The capital gains yield for the equity will be the total return (required return) minus the dividend yield. W: P0 = D0(1 + g) / (R – g) = £4.00(1.10)/(.15 – .10) = £88.00 Dividend yield = D1/P0 = £4.00(1.10)/ £88.00 = 5% Capital gains yield = .15 – .05 = 10% X: P0 = D0(1 + g) / (R – g) = £4.00/(.15 – 0) = £26.67 Dividend yield = D1/P0 = £4.00/£26.67 = 15% Capital gains yield = .15 – .15 = 0% Y: P0 = D0(1 + g) / (R – g) = £4.00(1 – .05)/(.15 + .05) = £19 Dividend yield = D1/P0 = £4.00(0.95)/ £19 = 20% Capital gains yield = .15 – .20 = – 5% Z: P2 = D2(1 + g) / (R – g) = D0(1 + g1)2(1 + g2)/(R – g) = £4.00(1.20)2(1.12)/(.15 – .12) = £215.04 P0 = £4.00 (1.20) / (1.15) + £4.00 (1.20)2 / (1.15)2 + £215.04 / (1.15)2 = £171.13 Dividend yield = D1/P0 = £4.00(1.20)/£171.13 = 2.8% Capital gains yield = .15 – .028 = 12.2% In all cases, the required return is 15%, but the return is distributed differently between current income and capital gains. High-growth equities have an appreciable capital gains component but a relatively small current income yield; conversely, mature, negative- growth shares provide a high current income but also price depreciation over time. 35. a. Using the constant growth model, the price of the stock paying annual dividends will be: P0 = D0(1 + g) / (R – g) = €3.00(1.06)/(.14 – .06) = €39.75 b. If the company pays semi-annual dividends instead of annual dividends, the semi- annual dividend will be one-half of the annual dividend, or: Semi-annual dividend: €3.00(1.06)/2 = €1.59 To find the equivalent annual dividend, we must assume that the semi-annual dividends are reinvested at the required return. We can then use this interest rate to find the equivalent annual dividend. In other words, when we receive the semi-annual dividend, we reinvest it at the required return on the equity. So, the effective semi-annual rate is: Effective semi-annual rate: 1.14.5 – 1 = .0677 The effective annual dividend will be the FVA of the semi-annual dividend payments at the effective semi-annual required return. In this case, the effective annual dividend will be: Effective D1 = €1.59(FVIF6.77%,2) = €3.29 Now, we can use the constant growth model to find the current share price as: P0 = €3.29/(.14 – .06) = €41.10 Note that we can not simply find the semi-annual effective required return and growth rate to find the value of the equity. This would assume the dividends increased every six months, not each year. 36. In this problem, growth is occurring from two different sources: the learning curve and the new project. We need to separately compute the value from the two different sources. First, we will compute the value from the learning curve, which will increase at 5 percent. All earnings are paid out as dividends, so we find the earnings per share are: EPS = Earnings/total number of outstanding shares EPS = (£10,000,000 × 1.05) / 10,000,000 EPS = £1.05 From the NPVGO model: P = E/(k – g) + NPVGO P = £1.05/(0.10 – 0.05) + NPVGO P = £21 + NPVGO Now we can compute the NPVGO of the new project to be launched two years from now. The earnings per share two years from now will be: EPS2 = £1.00(1 + .05)2 EPS2 = £1.1025 Therefore, the initial investment in the new project will be: Initial investment = .20(£1.1025) Initial investment = £0.22 The earnings per share of the new project is a perpetuity, with an annual cash flow of: Increased EPS from project = £5,000,000 / 10,000,000 shares Increased EPS from project = £0.50 So, the value of all future earnings in year 2, one year before the company realizes the earnings, is: PV = £0.50 / .10 PV = £5.00 Now, we can find the NPVGO per share of the investment opportunity in year 2, which will be: NPVGO2 = –£0.22 + £5.00 NPVGO2 = £4.78 The value of the NPVGO today will be: NPVGO = £4.78 / (1 + .10)2 NPVGO = £3.95 Plugging in the NPVGO model we get; P = £21 + £3.95 P = £24.95 Note that you could also value the company and the project with the values given, and then divide the final answer by the shares outstanding. The final answer would be the same. 37. Use the dividend growth model (P0 = Div1/(r-g) to find the discount rate. Source Growth Rate Discount Rate DPS growth (5 yr) 9.52% 12.94% EPS growth (5 yr) 16.85% 20.50% EPS growth (1 yr) 16.01% 19.63% DPS growth (1 yr) 17.98% 21.66% All, with the exception of the 5-year DPS growth rate seem very high. You can also calculate your own growth rate given that you know the ROE and can estimate the retention rate. This is good for a comparison: g = b x ROE = (1 – (1.78/7.96)) x 18.66% = .1449 = 14.49% With g = 14.49%, you get a discount rate of 18.06%. With a range of discount rates, you should allow the student to try and value Michelin using their own estimate of discount rate and growth rate. There is no wrong answer here, except if the wrong formula has been used! 38. c 6% Semi -annual Coupon 3% t 10 Semi-annual YTM 5.830% YTM 12% Face Value 100 Price £67.09 c 0.06 Semi Annual Coupon =(C6/2)*C9 t 10 Semi-annual YTM =(1+C8)^0.5-1 YTM 0.12 Face Value 100 Price =PV(F7,C7*2,-F6)+C9/(1+F7)^(C7*2) Chapter 6 Net Present Value and Other Investment Rules 1. The main strengths of NPV are that it a) takes into account the time value of money, b) incorporates scale in the estimate, and c) considers all cash flows. NPV also has the value additivity property and uses actual cash flows instead of accounting figures. The main weakness of NPV is that it requires the estimation of a discount rate. In addition, it provides a point estimate of the value of an investment, instead of a confidence interval. 2. Payback period is simply the accounting break-even point of a series of cash flows. To actually compute the payback period, it is assumed that any cash flow occurring during a given period is realized continuously throughout the period, and not at a single point in time. The payback is then the point in time for the series of cash flows when the initial cash outlays are fully recovered. Given some predetermined cutoff for the payback period, the decision rule is to accept projects that pay back before this cutoff, and reject projects that take longer to pay back. The worst problem associated with the payback period is that it ignores the time value of money. In addition, the selection of a hurdle point for the payback period is an arbitrary exercise that lacks any steadfast rule or method. The payback period is biased towards short-term projects; it fully ignores any cash flows that occur after the cutoff point. The main strength of payback period is its simplicity. 3. Obviously, discounted payback period incorporates the time value of money, which is an improvement over conventional payback period. However, if a manager has estimated an appropriate discount rate, she should use NPV analysis given its superiority as a method. One possible reason for why a manager may use discounted payback period is its simplicity. However, this does not make much sense given that the most difficult part of a present value analysis is the estimation of the discount rate. 4. The average accounting return is interpreted as an average measure of the accounting performance of a project over time, computed as some average profit measure attributable to the project divided by some average balance sheet value for the project. This text computes AAR as average net income with respect to average (total) book value. Given some predetermined cutoff for AAR, the decision rule is to accept projects with an AAR in excess of the target measure, and reject all other projects. AAR is not a measure of cash flows or market value, but is rather a measure of financial statement accounts that often bear little resemblance to the relevant value of a project. In addition, the selection of a cutoff is arbitrary, and the time value of money is ignored. For a financial manager, both the reliance on accounting numbers rather than relevant market data and the exclusion of time value of money considerations are troubling. Despite these problems, AAR continues to be used in practice because (1) the accounting information is usually available, (2) analysts often use accounting ratios to analyze firm performance, and (3) managerial compensation is often tied to the attainment of target accounting ratio goals. 5. NPV is theoretically the most correct method when assessing the viability of an investment. This is because NPV takes into account the scale of cash flows whereas IRR is a return measure and lacks scale. The reason why IRR is more popular in many countries is because of its simplicity and intuitive interpretation. 6. There are several issues the analyst must consider when using IRR to value investments. First, they must consider whether a collection of candidate projects are mutually exclusive and/or independent. It is not possible to compare projects if the scale of their investments are different because a small scale project may have a high IRR but actually increase wealth by very little in absolute terms. One must also be careful when using IRR for projects that require investment or borrowing. If the cash flows from the project change sign more than once, then it is very likely that the project will have more than one IRR value. 7. The profitability index is the present value of cash inflows relative to the project cost. As such, it is a benefit/cost ratio, providing a measure of the relative profitability of a project. The profitability index decision rule is to accept projects with a PI greater than one, and to reject projects with a PI less than one. The profitability index can be expressed as: PI = (NPV + cost)/cost = 1 + (NPV/cost). If a firm has a basket of positive NPV projects and is subject to capital rationing, PI may provide a good ranking measure of the projects, indicating the “bang for the buck” of each particular project. It’s main weaknesses are that it doesn’t take into account the size of cash flows because it is a ratio. 8. It is a puzzle why more companies do not use NPV. Most companies will draw on a variety of capital budgeting techniques rather than rely on just one. In this regard, payback period and IRR will be used as backup techniques to NPV. In addition, payback period and IRR can be cost effective when the size of investment is small. It is not particularly sensible to use the more complex NPV method of valuing projects when the net absolute return is minimal compared to the size of the firm. 9. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal the initial investment. Project A: Cumulative cash flows Year 1 = £4,000 = £4,000 Cumulative cash flows Year 2 = £4,000 +£5,500 = £9,500 Cumulative cash flows Year 3 = £4,000 + £5,500 + £8,000 = £17,500 > £14,000 The payback period is more than the 2 year cut-off and so project A would be rejected. Project B: Cumulative cash flows Year 1 = £4,500 = £4,500 Cumulative cash flows Year 2 = £4,500 + £2,200 = £6,700 > £6,000 The payback period is less than the 2 year cut-off and so project B would be accepted. Companies can calculate a more precise value using fractional years. To calculate the fractional payback period, find the fraction of year 2’s cash flows that is needed for the company to have cumulative undiscounted cash flows of £6,000. Divide the difference between the initial investment and the cumulative undiscounted cash flows as of year 2 by the undiscounted cash flow of year 2. Payback period = 1 + (£6,000 – £4,500) / £2,200 Payback period = 1.68 years b. Discount each project’s cash flows at 12 percent. Choose the project with the highest NPV. Project A: NPV = –£14,000 + £4,000 / 1.12 + £5,500 / 1.122 + £8,000 / 1.123 NPV = -£349.76 Project B: NPV = –£6,000 + £4,500 / 1.12 + £2,200 / 1.122 + £200 / 1.123 NPV = -£85.96 The firm should choose neither project since both have negative NPVs. 10. Since payback period does not take into account time value of money and the sum of the cash flows exactly equals the initial investment, the NPV can only be zero with a zero discount rate. It can never be positive. A zero discount rate is exceptionally unlikely to occur in real life and so the NPV of the project must be negative. 11. Payback Period Cumulative cash flows Year 1 = €35,000 = €35,000 Cumulative cash flows Year 2 = €35,000 + €10,000 = €45,000 The fractional part of year 2 is €10,000/€15,000 = 0.67. The payback period is 1.67 years. The minimum discount rate to have an acceptable NPV is the internal rate of return of the project. This is calculated by trial and error: NPV = 0 = –€45,000 + €35,000 / (1 + IRR) + €15,000 / (1+IRR)2 + €5,000 / (1+IRR)3 IRR = 15.12% 12. When we use discounted payback, we need to find the value of all cash flows today. The value today of the project cash flows for the first four years is: Value today of Year 1 cash flow = £20,000/1.14 = £17,543.86 Value today of Year 2 cash flow = £35,400/1.142 = £27,239.15 Value today of Year 3 cash flow = £48,000/1.143 = £32,398.63 Value today of Year 4 cash flow = £54,500/1.144 = £32,268.38 To find the discounted payback, we use these values to find the payback period. The 3 year cumulative discounted cash flow is £17,543.86 + £27,239.15 + £32,398.63 = £77,181.64. The fractional part of the third year to make the discounted cash flows equal the initial investment is (£100,000 - £77,181.64)/£32,268.38 = .21. So discounted payback period is 3.21 years. Since discounted cash flows sum to £109,450.02 there is no payback period when the initial cost is £120,000 or £170,000. 13. a. The average accounting return is the average project earnings after taxes, divided by the average book value, or average net investment, of the machine during its life. The book value of the machine is the gross investment minus the accumulated depreciation. Average book value = (Book value0 + Book value1 + Book value2 + Book value3 + Book value4 + Book value5) / (Economic life) Average book value = (€16,000 + 12,800 + 10,240 + 8,192 + 6,553.6) / (5 years) Average book value = €10,757.12 Average project earnings = €4,500 To find the average accounting return, we divide the average project earnings by the average book value of the machine to calculate the average accounting return. Doing so, we find: Average accounting return = Average project earnings / Average book value Average accounting return = €4,500 / €10,757.12 Average accounting return = 0.4183 or 41.83% b. The three flaws are that it doesn’t use cash flows, it doesn’t take into account the scale of the cash flows and it doesn’t consider the time value of money. 14. First, we need to determine the average book value of the project. The book value is the gross investment minus accumulated depreciation. Purchase Date Year 1 Year 2 Year 3 Gross Investment £8,000 €8,000 €8,000 €8,000 Less: Accumulated depreciation 0 2,667 5,333 8,000 Net Investment £8,000 5,333 2,667 €0 Now, we can calculate the average book value as: Average book value = (£8,000 + 5,333 + 2,667 + 0) / (4 years) Average book value = £4,000 To calculate the average accounting return, we must remember to use the after-tax average net income when calculating the average accounting return. So, the average aftertax net income is: Average after-tax net income = (1 – tc) Annual pre-tax net income Average after-tax net income = (1 – 0.23) £2,000 Average after-tax net income = £1,540 The average accounting return is the average after-tax net income divided by the average book value, which is: Average accounting return = £1,540 / £4,000 Average accounting return = 0.385 or 38.5% 15. When the discount rate is 0%, the NPV is equal to: When the discount rate is 50%, the NPV is equal to: When the discount rate is 100%, the NPV is equal to: The IRR is equal to the discount rate at which NPV=0: Through trial and error or Solver in an excel spreadsheet, IRR is equal to 142.68%. 16. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this Project A is: NPV =-R77,000 + R55,000 (1+0) + R320,000 (1+0)2 = R298,000 NPV =-R77,000 + R55,000 (1+0.5) + R320,000 (1+0.5)2 = R101,888.89 NPV =-R77,000 + R55,000 (1+1) + R320,000 (1+1)2 = R30,500 NPV =0 =-R77,000 + R55,000 (1+IRR) + R320,000 (1+IRR)2 NPV =0 =-D15,000 + D6,000 (1+IRR) + D7,500 (1+IRR)2 + D9,000 (1+IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 21.65% And the IRR for Project B is: Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR =21.90% 17. The profitability index is defined as the PV of the cash inflows divided by the PV of the cash outflows. The cash flows from this project are an annuity, so the equation for the profitability index is: PI = C(PVIFAR,t) / C0 PI = €500,000(PVIFA12%,7) / €2,300,000 PI = 0.9921 Since the PI is less than one, the project should be rejected. 18. a. The profitability index is the present value of the future cash flows divided by the initial cost. So, for Project Alpha, the profitability index is: PIAlpha = [€300 / 1.10 + €700 / 1.102 + €600 / 1.103] / €500 = 2.604 And for Project Beta the profitability index is: PIBeta = [€300 / 1.10 + €1,800 / 1.102 + €1,700 / 1.103] / €2,000 = 1.519 b. According to the profitability index, you would accept Project Alpha. However, remember the profitability index rule can lead to an incorrect decision when ranking mutually exclusive projects. 19. a. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this project is: 0 = C0 + C1 / (1 + IRR) + C2 / (1 + IRR)2 + C3 / (1 + IRR)3 + C4 / (1 + IRR)4 0 = £5,000 – £2,500 / (1 + IRR) – £2,000 / (1 + IRR)2 – £1,000 / (1 + IRR)3 – £1,000 / (1 +IRR)4 2 3 2,500 1,000 500 0 3,000 (1 ) (1 ) (1 ) D D D NPV D = = − + + IRR + + IRR + + IRR Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 13.99% b. This problem differs from previous ones because the initial cash flow is positive and all future cash flows are negative. In other words, this is a financing-type project, while previous projects were investing-type projects. For financing situations, accept the project when the IRR is less than the discount rate. Reject the project when the IRR is greater than the discount rate. IRR = 13.99% Discount Rate = 10% IRR > Discount Rate Reject the offer when the discount rate is less than the IRR. c. Using the same reason as part b., we would accept the project if the discount rate is 20 percent. IRR = 16.69% Discount Rate = 20% IRR < Discount Rate Accept the offer when the discount rate is greater than the IRR. d. The NPV is the sum of the present value of all cash flows, so the NPV of the project if the discount rate is 10 percent will be: NPV = £5,000 – £2,500 / 1.10 – £2,000 / 1.102 – £1,000 / 1.103 – £1,000 / 1.104 NPV = –£359.95 When the discount rate is 10 percent, the NPV of the offer is –£359.95. Reject the offer. And the NPV of the project is the discount rate is 20 percent will be: NPV = £5,000 – £2,500 / 1.2 – £2,000 / 1.22 – £1,000 / 1.23 – £1,000 / 1.24 NPV = £466.82 When the discount rate is 20 percent, the NPV of the offer is £466.82. Accept the offer. e. Yes, the decisions under the NPV rule are consistent with the choices made under the IRR rule since the signs of the cash flows change only once. 20. a. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR for each project is: Deepwater Fishing IRR: Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 24.30% Submarine Ride IRR: Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 21.46% Based on the IRR rule, the deepwater fishing project should be chosen because it has the higher IRR. b. To calculate the incremental IRR, we subtract the smaller project’s cash flows from the larger project’s cash flows. In this case, we subtract the deepwater fishing cash flows from the submarine ride cash flows. The incremental IRR is the IRR of these incremental cash flows. So, the incremental cash flows of the submarine ride are: Setting the present value of these incremental cash flows equal to zero, we find the incremental IRR is: Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: Incremental IRR = 19.92% For investing-type projects, accept the larger project when the incremental IRR is greater than the discount rate. Since the incremental IRR, 19.92%, is greater than the required rate of return of 15 percent, choose the submarine ride project. Note that this is the choice when evaluating only the IRR of each project. The IRR decision rule is flawed because there is a scale problem. That is, the submarine ride has a greater initial 2 3 €270,000 €350,000 €300,000 0 €600,000 NPV = = − + (1+ IRR) + (1+ IRR) + (1+ IRR) 2 3 €1,000,000 €700,000 €900,000 0 €1,800,000 NPV = = − + (1+ IRR) + (1+ IRR) + (1+ IRR) 2 3 €730,000 €350,000 €600,000 0 €1,200,000 NPV = = − + (1+ IRR) + (1+ IRR) + (1+ IRR) 0 1 2 3 Deepwater CF -€600,000 €270,000 €350,000 €300,000 Submarine CF -€1,800,000 €1,000,000 €700,000 €900,000 Incremental CF -€1,200,000 €730,000 €350,000 €600,000 investment than does the deepwater fishing project. This problem is corrected by calculating the IRR of the incremental cash flows, or by evaluating the NPV of each project. c. The NPV is the sum of the present value of the cash flows from the project, so the NPV of each project will be: Deepwater fishing: Submarine ride: Since the NPV of the submarine ride project is greater than the NPV of the deepwater fishing project, choose the submarine ride project. The incremental IRR rule is always consistent with the NPV rule. 21. a. The cumulative cash flows for the project are presented below: The payback period is equal to 4 + (1,000/6,000) = 4.1667 years b. The NPV is equal to: c. Since there are three changes of sign in the cash flows, there will be three IRRs. Using trial and error or solver, the only one that makes sense is: IRR = 10.92% d. The Profitability Index (PI) is equal to: PI = PV of cash flows subsequent to initial investment ÷ Initial investment = £15,393.51 / £15,000 =1.026 22. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal the initial investment. Trading Card game: 2 3 €270,000 €350,000 €300,000 €600,000 €96,687.76 NPV = − + (1+0.15) + (1+0.15) + (1+0.15) = 2 3 €1,000,000 €700,000 €900,000 €1,800,000 €190,630.39 NPV = − + (1+0.15) + (1+0.15) + (1+0.15) = 2 3 4 5 £4,000 £5,000 £2,000 £7,000 £7,000 £15,000 £393.51 NPV = − + (1+0.10) + (1+0.10) − (1+0.10) + (1+0.10) + (1+0.10) = 0 1 2 3 4 5 Cash Flow -£15,000 £4,000 £5,000 -£2,000 £7,000 £7,000 Cumulative CF -£15,000 -£11,000 -£6,000 -£8,000 -£1,000 £6,000 Payback period = €200 / €300 = .667 year Board Game: Payback period = €2,000/€2,200 = 0.909 year Since the Trading Card game has a shorter payback period than the Board Game project, the company should choose the Trading Card game. b. The NPV is the sum of the present value of the cash flows from the project, so the NPV of each project will be: Trading Card game: Board Game: Since the NPV of the Board Game is greater than the NPV of the Trading Card game, choose the Board Game. c. The IRR is the interest rate that makes the NPV of a project equal to zero. So, the IRR of each project is: Trading Card game: Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 90.13% Board Game: Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 50.83% Since the IRR of the Trading Card game is greater than the IRR of the Board Game, IRR implies we choose the trading Card game. d. To calculate the incremental IRR, we subtract the smaller project’s cash flows from the larger project’s cash flows. In this case, we subtract the Trading Card Game cash flows 2 3 €300 €100 €100 €200 €230.50 NPV = − + (1+0.10) + (1+0.10) + (1+0.10) = 2 3 €2,200 €900 €500 €2,000 €1,119.46 NPV = − + (1+0.10) + (1+0.10) + (1+0.10) = 2 3 €300 €100 €100 0 €200 NPV = = − + (1+ IRR) + (1+ IRR) + (1+ IRR) 2 3 €2,200 €900 €500 0 €2,000 NPV = = − + (1+ IRR) + (1+ IRR) + (1+ IRR) from the Board Game cash flows. The incremental IRR is the IRR of these incremental cash flows. So, the incremental cash flows are: Setting the present value of these incremental cash flows equal to zero, we find the incremental IRR is: Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: Incremental IRR = 46.31% For investing-type projects, accept the larger project when the incremental IRR is greater than the discount rate. Since the incremental IRR, 46.31%, is greater than the required rate of return of 10 per cent, choose the Board Game project. Note that this is the choice when evaluating only the IRR of each project. The IRR decision rule is flawed because there is a scale problem. That is, the Board Game has a greater initial investment than does the Trading Card game. This problem is corrected by calculating the IRR of the incremental cash flows, or by evaluating the NPV of each project. 23. The cash flows from the investment are as follows 0 1 2 3 4 5 Cash Flow -£220,000 £80,000 £84,000 £88,200 £92,610 £97,240.5 The NPV of the investment is: The IRR of the investment is: Using trial and error or solver, the IRR is: IRR = 27.69% 2 3 €1,900 €800 €400 0 €1,800 NPV = = − + (1+ IRR) + (1+ IRR) + (1+ IRR) 2 3 4 5 £80,000 £84,000 £88,200 £92,610 £97,240.5 £220,000 £112,047 NPV = − + (1+0.10) + (1+0.10) + (1+0.10) + (1+0.10) + (1+0.10) = 2 3 4 5 £80,000 £84,000 £88,200 £92,610 £97,240.5 0 £220,000 NPV = = − + (1+ IRR) + (1+ IRR) + (1+ IRR) + (1+ IRR) + (1+ IRR) 0 1 2 3 Trading Card CF -€200 €300 €100 €100 Board CF -€2,000 €2,200 €900 €500 Incremental CF -€1,800 €1,900 €800 €400 The Profitability Index of the Investment is: As NPV is positive, IRR is greater than the discounting rate and PI is greater than one, the revision should be undertaken. 24. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal the initial investment. People Carrier: Payback period = €200000 / €300000 = .667 year SUV: Payback period = 1+ €200,000/€250,000 = 1.8 years Since the People Carrier has a shorter payback period than the SUV, the company should choose the People Carrier. b. The NPV is the sum of the present value of the cash flows from the project, so the NPV of each project will be: People Carrier: SUV: Since the NPV of the People Carrier is greater than the NPV of the SUV, choose the People Carrier. c. The IRR is the interest rate that makes the NPV of a project equal to zero. So, the IRR of each project is: People Carrier: Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 90.13% SUV: 2 3 4 5 £80,000 £84,000 £88,200 £92,610 £97,240.5 / £220,000 (1 .10) (1 .10) (1 .10) (1 .10) (1 .10) 1.509 PI PI = + + + + + + + + + = 2 3 €300,000 €100,000 €100,000 €200,000 €218,754.56 NPV = − + (1+0.12) + (1+0.12) + (1+0.12) = 2 3 €300,000 €250,000 €250,000 €500,000 €145,100.67 NPV = − + (1+0.12) + (1+0.12) + (1+0.12) = 2 3 €300,000 €100,000 €100,000 0 €200,000 NPV = = − + (1+ IRR) + (1+ IRR) + (1+ IRR) Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 28.89% Since the IRR of the People Carrier is greater than the IRR of the SUV, IRR implies we choose the People Carrier. d. You do not need to calculate the incremental IRR since in this case because, even though the initial investment of the People Carrier is less, its NPV is greater than the SUV. 25. a. The profitability index is the PV of the future cash flows divided by the initial investment. The profitability index for each project is: PIA = [€70,000 / 1.12 + €70,000 / 1.122] / €100,000 = 1.18 PIB = [€130,000 / 1.12 + €130,000 / 1.122] / €200,000 = 1.10 PIC = [€75,000 / 1.12 + €60,000 / 1.122] / €100,000 = 1.15 b. The NPV of each project is: NPVA = –€100,000 + €70,000 / 1.12 + €70,000 / 1.122 NPVA = €18,303.57 NPVB = –€200,000 + €130,000 / 1.12 + €130,000 / 1.122 NPVB = €19,706.63 NPVC = –€100,000 + €75,000 / 1.12 + €60,000 / 1.122 NPVC = €14,795.92 c. Accept projects A, B, and C. Since the projects are independent, accept all three projects because the respective profitability index of each is greater than one. d. Accept Project B. Since the Projects are mutually exclusive, choose the Project with the highest PI, while taking into account the scale of the Project. Because Projects A and C have the same initial investment, the problem of scale does not arise when comparing the profitability indices. Based on the profitability index rule, Project C can be eliminated because its PI is less than the PI of Project A. Because of the problem of 2 3 €300,000 €250,000 €250,000 0 €500,000 NPV = = − + (1+ IRR) + (1+ IRR) + (1+ IRR) scale, we cannot compare the PIs of Projects A and B. However, we can calculate the PI of the incremental cash flows of the two projects, which are: Project C0 C1 C2 B – A –€100,000 €60,000 €60,000 When calculating incremental cash flows, remember to subtract the cash flows of the project with the smaller initial cash outflow from those of the project with the larger initial cash outflow. This procedure insures that the incremental initial cash outflow will be negative. The incremental PI calculation is: PI(B – A) = [€60,000 / 1.12 + €60,000 / 1.122] / €100,000 PI(B – A) = 1.014 The company should accept Project B since the PI of the incremental cash flows is greater than one. e. Remember that the NPV is additive across projects. Since we can spend €300,000, we could take two of the projects. In this case, we should take the two projects with the highest NPVs, which are Project B and Project A. 26. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal the initial investment. Dry Prepreg: Payback period = 2 years Solvent Prepreg: Payback period = 1+ €200,000/€600,000 = 1.333 years Since the Solvent has a shorter payback period than the Dry, the company should choose the Solvent. b. The NPV is the sum of the present value of the cash flows from the project, so the NPV of each project will be: Dry: Solvent: Since the NPV of the Dry is greater than the NPV of the Solvent, choose the Dry. 2 3 €500,000 €300,000 €900,000 €800,000 €578.662.66 NPV = − + (1+0.10) + (1+0.10) + (1+0.10) = 2 3 €400,000 €600,000 €200,000 €600,000 €409,767.09 NPV = − + (1+0.10) + (1+0.10) + (1+0.10) = c. The IRR is the interest rate that makes the NPV of a project equal to zero. So, the IRR of each project is: Dry: Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 43.38% Solvent: Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 48.88% Since the IRR of the Solvent is greater than the IRR of the Dry, IRR implies we choose the Solvent. d. To calculate the incremental IRR, we subtract the smaller project’s cash flows from the larger project’s cash flows. In this case, we subtract the Solvent cash flows from the Dry cash flows. The incremental IRR is the IRR of these incremental cash flows. So, the incremental cash flows are: Setting the present value of these incremental cash flows equal to zero, we find the incremental IRR is: Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: Incremental IRR = 33.67% 2 3 €500,000 €300,000 €900,000 0 €800,000 NPV = = − + (1+ IRR) + (1+ IRR) + (1+ IRR) 2 3 €400,000 €600,000 €200,000 0 €600,000 NPV = = − + (1+ IRR) + (1+ IRR) + (1+ IRR) 2 3 €100,000 €300,000 €700,000 0 €200,000 NPV (1 IRR) (1 IRR) (1 IRR) − = = − + + + + + + 0 1 2 3 Dry CF -€800,000 €500,000 €300,000 €900,000 Solvent CF -€600,000 €400,000 €600,000 €200,000 Incremental CF -€200,000 €100,000 -€300,000 €700,000 For investing-type projects, accept the larger project when the incremental IRR is greater than the discount rate. Since the incremental IRR, 33.67%, is greater than the required rate of return of 10 percent, choose the Dry Prepreg. Note that this is the choice when evaluating only the IRR of each project. The IRR decision rule is flawed because there is a scale problem. That is, the Dry has a greater initial investment than does the Solvent. This problem is corrected by calculating the IRR of the incremental cash flows, or by evaluating the NPV of each project. 27. a. To have a payback equal to the project’s life, given C is a constant cash flow for N years: C = I/N b. To have a positive NPV, I I / (PVIFAR%, N). c. Benefits = C (PVIFAR%, N) = 2 × costs = 2I C = 2I / (PVIFAR%, N) 28. a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal the initial investment. Project A: Cumulative cash flows Year 1 = €50,000 = €50,000 Cumulative cash flows Year 2 = €50,000 + €100,000 = €150,000 Payback period = 2 years Project B: Cumulative cash flows Year 1 = €200,000 = €200,000 Payback period = 1 year Project C: Cumulative cash flows Year 1 = €100,000 = €100,000 Payback period = 1 year Project B and Project C have the same payback period, so the projects cannot be ranked. Regardless, the payback period does not necessarily rank projects correctly. b. The IRR is the interest rate that makes the NPV of the project equal to zero, so the IRR of each project is: Project A: 0 = –€150,000 + €50,000 / (1 + IRR) + €100,000 / (1 + IRR)2 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRA = 0.00% And the IRR of the Project B is: 0 = –€200,000 + €200,000 / (1 + IRR) + €111,000 / (1 + IRR)2 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRB = 39.72% And the IRR of the Project C is: 0 = –€100,000 + €100,000 / (1 + IRR) + €100,000 / (1 + IRR)2 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRC = 61.80% The IRR criteria implies accepting Project C. c. Project A can be excluded from the incremental IRR analysis. Since the project has a negative NPV, and an IRR less than its required return, the project is rejected. We need to calculate the incremental IRR between Project B and Project C. In calculating the incremental cash flows, we subtract the cash flows from the project with the smaller initial investment from the cash flows of the project with the large initial investment, so the incremental cash flows are: Year Incremental cash flow 0 –€100,000 1 100,000 2 11,000 Setting the present value of these incremental cash flows equal to zero, we find the incremental IRR is: 0 = –€100,000 + €100,000 / (1 + IRR) + €11,000 / (1 + IRR)2 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: Incremental IRR = 10.00% For investing-type projects, accept the larger project when the incremental IRR is greater than the discount rate. Since the incremental IRR, 10.00 per cent, is less than the required rate of return of 20 per cent, choose the Project C. d. The profitability index is the present value of all subsequent cash flows, divided by the initial investment. We need to discount the cash flows of each project by the required return of each project. The profitability index of each project is: PIA = [€50,000 / 1.10 + €100,000 / 1.102] / €150,000 PIA = 0.85 PIB = [€200,000 / 1.20 + €111,000 / 1.202] / €200,000 PIB = 1.22 PIC = [€100,000 / 1.20 + €100,000 / 1.202] / €100,000 PIC = 1.53 The PI criteria implies accepting Project C. e. We need to discount the cash flows of each project by the required return of each project. The NPV of each project is: NPVA = –€150,000 + €50,000 / 1.10 + €100,000 / 1.102 NPVA = –€21,900.83 NPVB = –€200,000 + €200,000 / 1.20 + €111,000 / 1.202 NPVB = €43,750.00 NPVC = –€100,000 + €100,000 / 1.20 + €100,000 / 1.202 NPVC = €52,777.78 The NPV criteria implies accepting Project C. 29. Any combination of the investment appraisal techniques that are given in this chapter can be used. The worksheet given below provides the values for them. 0 1 2 3 4 5 6 7 8 9 10 Savings 4 4 4 4 4 4 4 4 4 4 Investment -9 1 CF -9 4 4 4 4 4 4 4 4 4 5 PV -9 3.5087 3.0778 2.6998 2.3683 2.0774 1.8223 1.5985 1.4022 1.2300 1.3487 Payback 2.25 years r = 14% Discounted Payback 2.89 years NPV Skr12.13m IRR 43.364% PI 2.348 30. Given the seven-year payback, the worst case is that the payback occurs at the end of the seventh year. Thus, the worst case: NPV = –£483,000 + £483,000/1.127 = –£264,515 The best case has infinite cash flows beyond the payback point. Thus, the best-case NPV is infinite. 31. The equation for the IRR of the project is: 0 = –£504 + £2,862/(1 + IRR) – £6,070/(1 + IRR)2 + £5,700/(1 + IRR)3 – £2,000/(1 + IRR)4 Using Descartes rule of signs, from looking at the cash flows we know there are four IRRs for this project. Even with most computer spreadsheets, we have to do some trial and error. From trial and error, IRRs of 25%, 33.33%, 42.86%, and 66.67% are found. We would accept the project when the NPV is greater than zero. See for yourself that the NPV is greater than zero for required returns between 25% and 33.33% or between 42.86% and 66.67%. 32. a. Here the cash inflows of the project go on forever, which is a perpetuity. Unlike ordinary perpetuity cash flows, the cash flows here grow at a constant rate forever, which is a growing perpetuity. If you remember back to the chapter on equity valuation, we presented a formula for valuing an equity with constant growth in dividends. This formula is actually the formula for a growing perpetuity, so we can use it here. The PV of the future cash flows from the project is: PV of cash inflows = C1/(R – g) PV of cash inflows = €80,000/(.12 – .06) = €1,333,333.33 NPV is the PV of the inflows minus by the PV of the outflows, so the NPV is: NPV of the project = –€800,000 + €1,333,333 = €533,333 The NPV is positive, so we would accept the project. b. Here we want to know the minimum growth rate in cash flows necessary to accept the project. The minimum growth rate is the growth rate at which we would have a zero NPV. The equation for a zero NPV, using the equation for the PV of a growing perpetuity is: 0 = – €800,000 + €80,000/(.12 – g) Solving for g, we get: g = 2% 33. a. The project involves three cash flows: the initial investment, the annual cash inflows, and the abandonment costs. The mine will generate cash inflows over its 12-year economic life. To express the PV of the annual cash inflows, apply the growing annuity formula, discounted at the IRR and growing at ten per cent. PV(Cash Inflows) = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t} PV(Cash Inflows) = R1,000,000{[1/(IRR – .1)] – [1/(IRR – .1)] × [(1 + .1)/(1 + IRR)]12} At the end of 12 years, Moshi Mining will abandon the mine, incurring a R500,000 charge. Discounting the abandonment costs back 12 years at the IRR to express its present value, we get: PV(Abandonment) = C12 / (1 + IRR)12 PV(Abandonment) = –R500,000 / (1+IRR)12 So, the IRR equation for this project is: 0 = –R6,000,000 + R1,000,000{[1/(IRR – .1)] – [1/(IRR – .1)] × [(1 + .1)/(1 + IRR)]12} –R500,000 / (1+IRR)12 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 21.54% b. Yes. Since the mine’s IRR exceeds the required return of 10 percent, the mine should be opened. The correct decision rule for an investment-type project is to accept the project if the discount rate is above the IRR. Although it appears there is a sign change at the end of the project because of the abandonment costs, the last cash flow is actually positive because the operating cash in the last year. 34. Both John and Beverley are incorrect. Take Beverley’s viewpoint first. It is true that if you calculate the future value of all intermediate cash flows to the end of the project at the required return, then calculate the NPV of this future value and the initial investment, you will get the same NPV. However, NPV says nothing about reinvestment of intermediate cash flows. The NPV is the present value of the project cash flows. What is actually done with those cash flows once they are generated is not relevant. Put differently, the value of a project depends on the cash flows generated by the project, not on the future value of those cash flows. The fact that the reinvestment “works” only if you use the required return as the reinvestment rate is also irrelevant simply because reinvestment is not relevant in the first place to the value of the project. One caveat: Our discussion here assumes that the cash flows are truly available once they are generated, meaning that it is up to firm management to decide what to do with the cash flows. In certain cases, there may be a requirement that the cash flows be reinvested. For example, in international investing, a company may be required to reinvest the cash flows in the country in which they are generated and not “repatriate” the money. Such funds are said to be “blocked” and reinvestment becomes relevant because the cash flows are not truly available. We now move on to John’s opinion, which is also incorrect. It is true that if you calculate the future value of all intermediate cash flows to the end of the project at the IRR, then calculate the IRR of this future value and the initial investment, you will get the same IRR. However, as with Beverley, what is done with the cash flows once they are generated does not affect the IRR. Consider the following example: C0 C1 C2 IRR Project A –£100 £10 £110 10% Suppose this £100 is a deposit into a bank account. The IRR of the cash flows is 10 per cent. Does the IRR change if the Year 1 cash flow is reinvested in the account, or if it is withdrawn and spent on pizza? No. Finally, consider the yield to maturity calculation on a bond. If you think about it, the YTM is the IRR on the bond, but no mention of a reinvestment assumption for the bond coupons is suggested. The reason is that reinvestment is irrelevant to the YTM calculation; in the same way, reinvestment is irrelevant in the IRR calculation. Our caveat about blocked funds applies here as well. 35. The problem with the three methods is that they all measure different things using different data. This means that there could be differences in the recommendations given by them. 36. To answer this question, we need to examine the incremental cash flows. To make the projects equally attractive, Project Billion must have a larger initial investment. We know this because the subsequent cash flows from Project Billion are larger than the subsequent cash flows from Project Million. So, subtracting the Project Million cash flows from the Project Billion cash flows, we find the incremental cash flows are: Incremental Year cash flows 0 –I0 + £1,500 1 £300 2 £300 3 £500 Now we can find the present value of the subsequent incremental cash flows at the discount rate, 12 per cent. The present value of the incremental cash flows is: PV = £1,500 + £300 / 1.12 + £300 / 1.122 + £500 / 1.123 PV = £2,362.91 So, if I0 is greater than £2,362.91, the incremental cash flows will be negative. Since we are subtracting Project Million from Project Billion, this implies that for any value over £2,362.91 the NPV of Project Billion will be less than that of Project Million, so I0 must be less than £2,362.91. 37. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR of the project is: 0 = £20,000 – £26,000 / (1 + IRR) + £13,000 / (1 + IRR)2 Even though it appears there are two IRRs, a spreadsheet, financial calculator, or trial and error will not give an answer. The reason is that there is no real IRR for this set of cash flows. If you examine the IRR equation, what we are really doing is solving for the roots of the equation. Going back to high school algebra, in this problem we are solving a quadratic equation. In case you don’t remember, the quadratic equation is: x = In this case, the equation is: x = The square root term works out to be: 676,000,000 – 1,040,000,000 = –364,000,000 The square root of a negative number is a complex number, so there is no real number solution, meaning the project has no real IRR. a b b ac 2 − 2 − 4 ( 26,000) ( 26,000)2 4(20,000)(13,000) 2(20,000) Solution Manual for Corporate Finance David Hillier, Stephen Ross, Randolph Westerfield, Jeffrey Jaffe, Bradford Jordan 9780077139148
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